Spectral analysis of one-dimensional Dirac system with summable potential and Sturm–Liouville operator with distribution coefficients

We consider one-dimensional Dirac operator $\mathcal{L}_{P,U}$ with Birkhoff regular boundary conditions and summable potential $P(x)$ on $[0,\pi].$ We introduce strongly and weakly regular operators. In both cases, asymptotic formulas for eigenvalues are found. In these formulas, we obtain main asymptotic terms and estimates for the second term. We specify these estimates depending on the functional class of the potential: $L_p[0,\pi]$ with $p\in[1,2]$ and the Besov space $B_{p,p'}^\theta[0,\pi]$ with $p\in[1,2]$ and $\theta\in(0,1/p).$ Additionally, we prove that our estimates are uniform on balls $\|P\|_{p,\theta}\le R.$ Then we get asymptotic formulas for normalized eigenfunctions in the strongly regular case with the same residue estimates in uniform metric on $x\in[0,\pi].$ In the weakly regular case, the eigenvalues $\lambda_{2n}$ and $\lambda_{2n+1}$ are asymptotically close and we obtain similar estimates for two-dimensional Riesz projectors. Next, we prove the Riesz basis property in the space $(L_2[0,\pi])^2$ for a system of eigenfunctions and associated functions of an arbitrary strongly regular operator $\mathcal{L}_{P, U}.$ In case of weak regularity, the Riesz basicity of two-dimensional subspaces is proved.
In parallel with the $\mathcal{L}_{P,U}$ operator, we consider the Sturm–Liouville operator $\mathcal{L}_{q,U}$ generated by the differential expression $-y''+q(x)y$ with distribution potential $q$ of first-order singularity (i.e., we assume that the primitive $u=q^{(-1)}$ belongs to $L_2[0,\pi]$) and Birkhoff-regular boundary conditions. We reduce to this case operators of more general form $- (\tau_1y')'+i(\sigma y)'+i\sigma y'+\tau_0y,$ where $\tau_1', \sigma, \tau_0^{(-1)}\in L_2$ and $\tau_1>0.$ For operator $\mathcal{L}_{q,U},$ we get the same results on the asymptotics of eigenvalues, eigenfunctions, and basicity as for operator $\mathcal{L}_{P, U}.$
Then, for the Dirac operator $\mathcal{L}_{P, U},$ we prove that the Riesz basis constant is uniform over the balls $\|P\|_{p,\theta}\le R$ for $p>1$ or $\theta>0.$ The problem of conditional basicity is naturally generalized to the problem of equiconvergence of spectral decompositions in various metrics. We prove the result on equiconvergence by varying three indices: $f\in L_\mu[0,\pi]$ (decomposable function), $P\in L_\varkappa[0,\pi]$ (potential), and $\|S_m-S_m^0\|\to0,$ $m\to\infty,$ in $L_\nu[0,\pi]$ (equiconvergence of spectral decompositions in the corresponding norm). In conclusion, we prove theorems on conditional and unconditional basicity of the system of eigenfunctions and associated functions of operator $\mathcal{L}_{P, U}$ in the spaces $L_\mu[0,\pi],$ $\mu\ne2,$ and in various Besov spaces $B_{p, q}^\theta[0,\pi].$